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 A055417 Number of points in N^n of norm <= 2. 3
 1, 3, 6, 11, 20, 36, 63, 106, 171, 265, 396, 573, 806, 1106, 1485, 1956, 2533, 3231, 4066, 5055, 6216, 7568, 9131, 10926, 12975, 15301, 17928, 20881, 24186, 27870, 31961, 36488, 41481, 46971, 52990, 59571, 66748, 74556, 83031, 92210, 102131, 112833, 124356 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial transform of [1, 0, 2, -1, 2, -1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Mar 12 2009 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1000 FORMULA a(n) = (n^3 - 3*n^2 + 14*n + 24)*(n+1)/24. Proof: The coordinates of such a point are a permutation of one of the vectors (0, ..., 0), (0, ..., 0, 1), (0, ..., 0, 2), (0, ..., 0, 1, 1), (0, ..., 0, 1, 1, 1), or (0, ..., 0, 1, 1, 1, 1), so the number of points is 1 + n + n + binomial(n,2) + binomial(n,3) + binomial(n,4). - Formula conjectured by Frank Ellermann, Mar 16 2002 and explained by Michael Somos, Apr 25 2003 G.f.: (1-2*x+x^2+x^3)/(1-x)^5. - Michael Somos, Apr 25 2003 EXAMPLE {(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1), (0, 2, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1), (2, 0, 0)} are all the points in N^3 of norm <= 2 so a(3)=11. MATHEMATICA CoefficientList[Series[(-z^3 - z^2 + 2*z - 1)/(z - 1)^5, {z, 0, 100}], z] (* and *) Table[(n^4 - 6*n^3 + 23 n^2 + 6*n)/24, {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *) PROG (PARI) a(n)=(n^3-3*n^2+14*n+24)*(n+1)/24 CROSSREFS Row n=2 of A302998. Sequence in context: A208851 A182845 A265076 * A018918 A077855 A054887 Adjacent sequences: A055414 A055415 A055416 * A055418 A055419 A055420 KEYWORD nonn AUTHOR STATUS approved

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Last modified December 3 09:50 EST 2022. Contains 358517 sequences. (Running on oeis4.)